Magnetic Field Amplification in Galaxy Clusters and Its Simulation · 2018-12-18 · et al. 2010)....

Space Sci Rev (2018) 214:122https://doi.org/10.1007/s11214-018-0556-8

Magnetic Field Amplification in Galaxy Clusters and ItsSimulation

J. Donnert1,2 · F. Vazza3 · M. Brüggen4 · J. ZuHone5

Received: 3 July 2018 / Accepted: 30 October 2018 / Published online: 13 November 2018© The Author(s) 2018

Abstract We review the present theoretical and numerical understanding of magnetic fieldamplification in cosmic large-scale structure, on length scales of galaxy clusters and beyond.Structure formation drives compression and turbulence, which amplify tiny magnetic seedfields to the microGauss values that are observed in the intracluster medium. This processis intimately connected to the properties of turbulence and the microphysics of the intra-cluster medium. Additional roles are played by merger induced shocks that sweep throughthe intra-cluster medium and motions induced by sloshing cool cores. The accurate simula-tion of magnetic field amplification in clusters still poses a serious challenge for simulationsof cosmological structure formation. We review the current literature on cosmological sim-ulations that include magnetic fields and outline theoretical as well as numerical challenges.

Keywords Galaxy clusters · Magnetic fields · Simulations · Magnetic dynamo

Clusters of Galaxies: Physics and CosmologyEdited by Andrei Bykov, Jelle Kaastra, Marcus Brüggen, Maxim Markevitch, Maurizio Falanga andFrederik Bernard Stefan Paerels

B J. [email protected]; [email protected]

F. [email protected]

M. Brü[email protected]

J. [email protected]

1 via P. Gobetti 101, 40129 Bologna, Italy

2 Present address: Leiden Observatory, PO Box 9513, 2300 RA Leiden, The Netherlands

3 Dipartimento di Fisica e Astronomia, via Gobetti 93/2, 40129 Bologna, Italy

4 Hamburg Observatory, Gojenbergsweg 112, 21029 Hamburg, Germany

5 Smithsonian Astrophysical Observatory, 60 Garden St., Cambridge, MA 02138, USA

http://crossmark.crossref.org/dialog/?doi=10.1007/s11214-018-0556-8&domain=pdf

http://orcid.org/0000-0001-9627-2980

mailto:[email protected]





122 of 49 J. Donnert et al.

1 Introduction

Magnetic fields permeate our Universe, which is filled with ionized gas from the scales ofour solar system up to filaments and voids in the large-scale structure (Klein and Fletcher2015). While magnetic fields are usually not dynamically important, their presence shapesthe physical properties of the Baryonic medium (Schekochihin and Cowley 2007). On thelargest scales, radio observations remain our most important tool to estimate magnetic fieldstoday (see e.g. van Weeren, this volume). Recent and upcoming advances in instrumentationenable the observation of radio emission on scales of a few kpc at the cluster outskirts andwill soon provide three-dimensional magnetic field distributions in the inter-cluster-medium(ICM) through Faraday tomography (Govoni et al. 2014).

Connecting these new observations to theoretical expectations is a major challenge forthe community, due to the complexity of the non-thermal physics in the cosmological con-text. In the framework of cold Dark Matter, structure formation is dominated by gravitationalforces and proceeds from the bottom up: smaller DM halos form first (Planelles et al. 2016),and baryons flow into the resulting potential well. Through cooling, stars and galaxies formand evolve into larger structures (groups, clusters, filaments), by infall and merging (Moet al. 2010). These processes drive turbulent gas motions and a magnetic dynamo that am-plifies some form of seed field to μG values in the center of galaxy clusters. Galaxy feedbackinjects magnetic fields and relativistic particles (cosmic-ray protons and electrons) into thelarge-scale structure that interact with shocks and turbulence, get (re-)accelerated and finallybecome observable at radio frequencies and potentially in the γ -ray regime (Schlickeiser2002; Lazarian et al. 2012; Brunetti and Jones 2014).

In the past decade significant progress has been made in the simulation of galaxy for-mation, with an emphasis on physical models for feedback (e.g. Naab and Ostriker 2017).Unfortunately, the same is not true for the simulation of turbulence, magnetic fields andcosmic-ray evolution—nearly every step in the chain of non-thermal processes remains opentoday:What is the origin of the magnetic seed fields and the contributions of various astrophys-ical sources? What are the properties of turbulence and the magnetic dynamo in the ICM,filaments, and voids? What is the distribution and topology of magnetic fields? What is thespatial distribution of radio dark cosmic-ray electrons in clusters? Where are the cosmic-ray protons? What are their sources? What physics governs particle acceleration in shocksthat leads to radio relics? How does turbulence couple to cosmic-rays in radio halos? Whatare the physical properties (viscosity, resistivity, effective collisional scales) of the diffuseplasma in the ICM, filaments and voids?

Answers have proven themselves difficult to obtain, in part because turbulence is a de-manding numerical problem, but also because the physics is different enough from galaxyformation to make some powerful numerical approaches like density adaptivity rather in-effective. Today, JVLA and LOFAR observations have achieved an unprecedented spatialand spectral detail in the observation of magnetic phenomena in cluster outskirts (e.g. Owenet al. 2014; Hoang et al. 2017; Rajpurohit et al. 2018), thereby challenging simulations toincrease their level of spatial and physical detail. The gap will likely widen in the next yearsas SKA precursors like ASKAP see first light (Gaensler et al. 2010) and results from theLOFAR survey key science project become available (Shimwell et al. 2017).

Here we review the current status on astrophysical and cosmological simulations of mag-netic field amplification in structure formation through compression, shocks, turbulence andcosmic-rays. Such a review will naturally emphasize galaxy clusters, simply because thereis only weak observational evidence for magnetic fields in filaments and voids. We will

Magnetic Field Amplification in Galaxy Clusters and Its Simulation of 49 122

also touch on ideal MHD as a model for intergalactic plasmas and introduce fundamentalconcepts of turbulence and the MHD dynamo. We are putting an emphasis on numericalsimulations because they are our most powerful tool to study the interplay of non-thermalphysics. This must also include some details on common algorithms for MHD and their lim-itations. Today, these algorithms and their implementation limit our ability to model shocks,turbulence and the MHD dynamo in a cosmological framework.

We exclude from this review topics that are not directly related to simulations of thecosmic magnetic dynamo. While we shortly introduce turbulence and dynamo theory, wedo not attempt to go into detail, several reviews are available (e.g. Schekochihin and Cowley2007, for an introduction). We also do not review models for particle acceleration in clusters(Brunetti and Jones 2014) or observations (see Ferrari et al. 2008, and van Weeren et al., thisvolume). We also do not discuss in detail the seeding of magnetic fields (see Widrow et al.2012; Ryu et al. 2012; Subramanian 2016, for recent exhaustive reviews on the topic), northe amplification of magnetic fields in the interstellar medium (see Federrath 2016, for arecent review) or in galaxies (e.g. Schleicher et al. 2010; Beck et al. 2012; Martin-Alvarezet al. 2018, for theoretical reviews).

1.1 Overview

Galaxy clusters form through the gravitational collapse and subsequent merging of virializedstructures into haloes, containing about 80% Dark Matter and 20% Baryons (Sarazin 2002;Voit 2005; Planelles et al. 2015). From X-ray observations we know that the diffuse thermalgas in the center of haloes with masses > 1014 M� is completely ionized, with tempera-tures of T = 108 K and number densities of nth ≈ 10−3 cm−3, (e.g. Sarazin 1988; Borganiet al. 2008). The speed of sound is then cs = √

γP/ρ ≈ 1200 km/s, where γ = 5/3 is theadiabatic index at density ρ and pressure P .

The ideal equation of state for a monoatomic gas is applicable in such a hot under-densemedium, even though the ICM contains ≈ 25% helium and heavier elements as well (e.g.Böhringer and Werner 2010). In fact, the intracluster medium is one of the most ideal plas-mas known, with a plasma parameter of g ≈ 10−15 and a Debye length of λD ≈ 105 cm thatstill contains ≈ 1012 protons and electrons. In contrast, the mean free path for Coulomb col-lisions is in the kpc regime (Eq. (6)). Clearly, electromagnetic particle interactions dominateover two-body Coulomb collisions and plasma waves shape the properties of the mediumon small scales (e.g. Schlickeiser 2002, Table 8.1).

Cluster magnetic fields of 1 μG were first estimated from upper limits on the diffuse syn-chrotron emission of intergalactic material in a 1 Mpc3 volume by Burbidge (1958). Withthe discovery of the Coma radio halo by Willson (1970), this was confirmed using equiparti-tion arguments between the cosmic-ray electron energy density and magnetic energy density(e.g. Beck and Krause 2005). Later estimates based on the rotation measure of backgroundsources to the Coma cluster obtain central magnetic fields of 3–7 μG scaling with ICM ther-mal density with an exponent of 0.5–1 (e.g. Bonafede et al. 2010). Hence the ICM is a highβ = nthkBT/B2 ≈ 100 plasma, where thermal pressure dominates magnetic pressure.

Based on above estimates, one may hope that magneto-hydrodynamics (MHD) is appli-cable on large enough scales in clusters (Sects. 3.2.1 and 3.2.2). Then the magnetic field Bevolves with the flow velocity v according to the induction equation (Landau et al. 1961):

∂B∂t

= −v · ∇B + B · ∇v − B∇ · v + η�B, (1)

where the first term accounts for the advection of field lines, the second one for stretching,the third term for compression and the fourth term for the magnetic field dissipation with the


diffusivity η = cs/4πσ and the conductivity σ . Because the ICM is a nearly perfect plasma(βpl � 1), conductivity is very high, diffusivity likely very low (η ≈ 0). Then the inductionequation (1) predicts that magnetic fields are frozen into the plasma and advected with thebulk motions of the medium (Kulsrud and Ostriker 2006). Because Eq. (1) is a conserva-tion equation for magnetic flux, magnetic fields cannot be created in the MHD framework,but have to be seeded by some mechanism, also at high redshift (Sect. 2). However, currentupper limits on large-scale magnetic fields exclude large-scale seed fields above ∼ nG (Adeet al. 2016), and back-of-the-envelope calculations show that pure compression cannot pro-duce μG in clusters from such initial values (Sect. 3.1). X-ray observations have revealedsubstantial turbulent velocities in a few clusters (Schuecker et al. 2004; Zhuravleva et al.2014; Aharonian et al. 2016). These are in agreement with estimates from rotation measure-ments (Vogt and Enßlin 2003; Kuchar and Enßlin 2011) that can also be used to constrainmagnetic field power spectra (Vacca et al. 2012, 2016; Govoni et al. 2017).

It is reasonable to assume some form of turbulent dynamo in the clusters and possiblyfilaments (Jaffe 1980; Roland 1981; Ruzmaikin et al. 1989; De Young 1992; Goldshmidtand Rephaeli 1993; Kulsrud et al. 1997; Sánchez-Salcedo et al. 1998; Subramanian et al.2006; Enßlin and Vogt 2006), but it is necessary to consider plasma-physical arguments tounderstand the fast growth of seed fields by many orders of magnitude (Schekochihin et al.2005b; Schekochihin and Cowley 2007). There are clear theoretical predictions for idealizedMHD dynamos (e.g. Schekochihin et al. 2004; Porter et al. 2015), which show that magneticfields are amplified though an inverse cascade at the growing Alfvén scale, where the fieldstarts back-reacting on the flow. This is called the small-scale dynamo (Sect. 3.3). How-ever, the astrophysical situation differs significantly from these idealized models: structureformation drives turbulence localized, episodic and multi-scale in the presence of a stronggravitational potential in galaxy clusters (Sect. 3.5), and the magneto-hydrodynamical prop-erties of the medium are far from clear (Schekochihin et al. 2009). Shocks and cosmic-raysamplify magnetic fields as well and are very difficult to model (Sect. 5).

With JVLA, LOFAR, ASKAP and the SKA, the Alfvén scale comes within the rangeof radio observations: radio relics are now spatially resolved to a few kpc in polarization;low-frequency surveys are expected to find hundreds of radio halos and mini-halos; Faradaytomography will allow to map magnetic field structure also along the line of sight (see vanWeeren et al., this volume). Future X-ray missions will put stringent bounds on turbulentvelocities in clusters and constrain magnetic field amplification by draping and sloshing incold fronts (Sect. 6).

2 Magnetic Seeding Processes

Let us begin with a short overview of proposed seeding mechanisms; a detailed review canbe found e.g. in Subramanian (2016). It is very likely that more than one of these mech-anisms contributes to the magnetization of the large-scale structure. Hence an importantquestion for simulations of magnetic field amplification is the influence of these seedingmechanisms on the final magnetic field.

2.1 Primordial Mechanisms

Several mechanisms for the initial seed field have been suggested to start the dynamo ampli-fication process within galaxies and galaxy clusters. Some of the proposed scenarios involvethe generation of currents during inflation, phase transitions and baryogenesis (e.g. Harrison


1973; Kahniashvili et al. 2010, 2011, 2016; Widrow et al. 2012; Durrer and Neronov 2013;Subramanian 2016). These primordial seed fields may either produce small (≤ Mpc, e.g.Chernin 1967) or large (e.g. Zel’dovich 1970; Turner and Widrow 1988) coherence lengths,whose structure may still persist until today (e.g. Hutschenreuter et al. 2018), in the emptiestcosmic regions, possibly also carrying information on the generation of primordial helicity(e.g. Semikoz and Sokoloff 2005; Campanelli 2009; Kahniashvili et al. 2016).

Owing to uncertainties in the physics of high energy regimes in the early Universe, theuncertainty in the outcome of most of the above scenarios is rather large and fields in therange of ∼ 10−34–10−10 G are still possible.

The presence of magnetic fields with rms values larger than a few co-moving ∼ nG on≤ Mpc scales at z ≈ 1100 is presently excluded by the analysis of the CMB angular powerspectrum by Planck (Ade et al. 2016; Trivedi et al. 2014), while higher limits are derivedfor primordial fields with much larger coherence length (Barrow et al. 1997). Conversely,the lack of detected Inverse Compton cascade around high redshift blazars was used to setlower limits1 on cosmological seed fields of ≥ 10−16 G on ∼ Mpc (Dolag et al. 2009, 2011;Neronov and Vovk 2010; Arlen et al. 2014; Caprini and Gabici 2015; Chen et al. 2015).

2.2 Seeding from Galactic Outflows

At lower redshift (z ≤ 6) galactic feedback can transport magnetic fields from galactic tomore rarefied scales such as galaxy clusters. In lower mass haloes, star formation driveswinds of magnetized plasma into the circum-galactic medium (e.g. Kronberg et al. 1999;Völk and Atoyan 2000; Donnert et al. 2009; Bertone et al. 2006; Samui et al. 2017) and intovoids (Beck et al. 2013b). At the high mass end, active galactic nuclei (AGN) can magnetizethe central volume of clusters through jets (e.g. Dubois and Teyssier 2008; Xu et al. 2009;Donnert et al. 2009) and even the intergalactic medium during their violent quasar phase(Furlanetto and Loeb 2001). Just taking into account the magnetization from dwarf galaxiesin voids, a lower limit of the magnetic field in voids has been derived as ∼ 10−15 G (Becket al. 2013b; Samui et al. 2017).

If magnetic fields have been released by processes triggered during galaxy formation,they might have affected the transport of heat, entropy, metals and cosmic rays in formingcosmic structures (e.g. Planelles et al. 2016; Schekochihin et al. 2008).

Additional processes such as the “Biermann-battery” mechanism (Kulsrud et al. 1997),aperiodic plasma fluctuations in the inter-galactic plasma (Schlickeiser et al. 2012), resistivemechanisms (Miniati and Bell 2011) or ionization fronts around the first stars (Langer et al.2005) might provide additional amplification to the primordial fields starting from z ≤ 103,i.e. after recombination.

3 Magnetic Field Amplification in the Intra-cluster Medium

3.1 Amplification by Compression

From the third term in the induction equation (Eq. (1)) we find that a positive divergence ofthe velocity field ∇ · v, i.e. a net inflow, results in the growth of the magnetic field (Sur et al.

1See however Broderick et al. (2012) for a different interpretation.


Fig. 1 Magnetic field strength asa function of over density incosmological SPH simulations.Starting from 3 differentcosmological seed field strengths:2 × 10−13 G (dark green),2 × 10−12 G (black),8 × 10−12 G (green) (Dolaget al. 2008, 2005a). Adiabaticevolution solely by compressionin grey. Runs with galactic seedsare in red and blue

2012). Indeed, it is a basic result of MHD that magnetic flux Φ is conserved (e.g. Kulsrudand Ostriker 2006) leading to the scaling of the magnetic field with density:

B(ρ) ∝ B(z�)

(ρ

〈ρ〉)2/3

. (2)

For a galaxy cluster with an average over-density of � = ρ/〈ρ〉 ≈ 100 this means that adia-batic compression can amplify the seed field by up to a factor of ∼ 20 within the virial radius(or ∼ 180 within the cluster core, where the density can be ≈ 2500 the mean density). Thisrefers to the average magnetic field inside a radius of the cluster. The peak density and mag-netic field can be much higher. However, depending on the redshift and environment of theseed fields, the expectation from adiabatic amplification can be lower. Nonetheless, observa-tions find a scaling exponent of magnetic field strength with cluster density of 0.5–1, whichis compatible with amplification by compression.

In Fig. 1 we reproduce a central result from early cosmological SPMHD (smooth particlemagneto-hydrodynamics) simulations (Dolag et al. 2005a, 2008). They show the magneticfield strength over density in a cosmological simulation with cosmological seed fields ofB(z�) = 2 × 10−13 G (dark green), B(z�) = 2 × 10−12 G (black), B(z�) = 8 × 10−12 G (darkgreen) co-moving, seeded at z� = 20 alongside the analytical expectation from Eq. (2). Runswith galactic seeding in blue and red. At central cluster over-densities (ρ/〈ρ〉 > 1000),all but one simulations reach μG field strengths. Thus different seeding models are in-distinguishable here. Differences to galactic field seeding appear only at lower densi-ties.

In runs with a cosmological seed field, amplification is mostly caused by compressionbelow over-densities of 1000. At larger over-densities, a dynamo caused by velocity gradi-ents along the field lines in the first term of the induction equation (1) operates and leadsto much higher field strengths. This is characteristic for turbulence in structure formation,which we will discuss next. Simulations of the cosmic dynamo and their limitations will becovered later in Sect. 4.


3.2 A Brief Introduction to Turbulence

Let us first introduce a few key concepts of turbulence used throughout the review. For amore detailed exposure, we refer the reader to the vast literature available on Astrophysicalturbulence (e.g. Landau and Lifshitz 1966; Kulsrud and Ostriker 2006; Lazarian et al. 2009).

A key idea of the Kolmogorov picture of turbulence is that random fluid motions withvelocity dispersion2 v of size or scale l (“eddies”) break up into two eddies of half the sizedue to the convective v · ∇v term in the fluid equations. This process constitutes a localenergy transfer from large to small scales at a rate kv, where k = 2π/l is the wave vector.This process continues at each smaller length scale which leads to a cascade of velocityfluctuations down to smaller scales with decreasing kinetic energy. At an inner scale kν ,the local kinetic energy becomes comparable to viscous forces, which dissipate the motioninto thermal energy or, in case of a dynamo, also magnetic energy via the Lorentz force. Ateach scale, the cascading time scale is the eddy turnover time τl = l/vl and for continuousinjection of velocity fluctuations at the outer scale a steady state is reached. If the kineticenergy density of these fluctuations is 1/2ρv2 = ρ/2

∫I (k)dk (assuming isotropy), then it

can be shown that the velocity power spectrum I (k) is (Kolmogorov 1941, 1991):

I (k) ∝ v20

k2/30

k5/3, (3)

where v0 is the velocity dispersion of the largest eddy at scale k0. We note that v0 is avelocity fluctuation on top of the mean. This dispersion of the associated random velocityfield then scales as v2 ∝ l2/3. It follows that the energy of turbulence is dominated by thelargest scales and that viscous forces are important close to the dissipative inner scale, wheremotions are slowest. The range of scales where Eq. (3) is valid is called the inertial range,and the Reynolds number is defined as:

Re = v0

k0ν(4)

∝(

l0

lν

)4/3

(5)

with the kinematic viscosity ν. The role of small scales is universal in the sense that thecascading does not depend on the driving scale or velocity (assuming homogeneity, scaleinvariance, isotropy and locality of interactions) (Schekochihin and Cowley 2007). We notethat turbulence is not limited to velocity fluctuations around a mean caused by a superposi-tion of velocity eddies. The velocity field causes density and pressure fluctuations as well,because these are coupled via the fluid equations. For sub-sonic turbulence the fluctuationswill be adiabatic. This has been used to place an upper limit on the kinematic viscosity inthe Coma cluster of ν < 3 × 1029 cm2/s on scales of 90 kpc using X-ray data (Schueckeret al. 2004).

Whether or not the stage of the dynamo amplification is reached in an astrophysicalsystem ultimately depends on the magnetic Reynolds number (Eq. (15)) and on the natureof the turbulent forcing in the ICM (Federrath et al. 2014; Beresnyak and Miniati 2016). Themagnetic Reynolds number is set by the outer scale and the dissipation scale, so it is worth

2Note that velocity and velocity dispersion (i.e. root-mean-square of the power-spectrum at scale k) areused somewhat interchangeably in the literature. Similarly we denote the velocity dispersion with v as thedistinction is usually clear by context.


discussing the latter next. For galaxy clusters, these scales are connected to the physics ofthe ICM plasma.

3.2.1 The Spitzer Model for the ICM

As noted in the introduction, most theoretical and numerical studies approximate the ICMplasma as a fluid. However, the MHD equations as a statistical description of the many-body plasma are applicable only, if equilibration processes between ionized particles acton length and time scales much smaller than “the scales of interest” of the fluid problem,i.e. if collisional equilibrium among particles (protons, electron, metal ions) is maintainedso local particle distributions become Maxwellian and temperature and pressure are welldefined (Landau and Lifshitz 1966).

In the “classic” physical picture of the ICM, this arises from ion-ion Coulomb scattering,with a viscosity νii, over a mean free path lmfp which is given by the Spitzer model for fullyionized plasmas (Spitzer 1956). It can be shown that a whole cluster is then “collisional” inthe sense that rvir � lmfp (Sarazin 1986), with:

lmfp ≈ 23

(nth

10−3 cm−3

)−1(T

108 K

)2

kpc. (6)

Under these conditions, the Reynolds number (Eq. (5)) of the ICM in a cluster during e.g. amajor merger is (e.g. Brunetti and Lazarian 2007):

Re = LvL

νii(7)

≈ 52vL

103 km/s· L

300 kpc· n

10−3 cm−3·(

T

8 keV

)−5/2

·(

logΛ

40

)(8)

where L is a typical eddy size (ideally the injection scale of turbulence), logΛ is theCoulomb logarithm (Longair 2011) and vL is the rms velocity within the scale L. Thusbased on typical values of the ICM, the Reynolds number would hardly reach Re ∼ 102 inmost conditions.

In contrast, rotation measures inferred from observations of radio galaxies have demon-strated field reversals on kpc scales, implying much larger Reynolds numbers (Laing et al.2008; Govoni et al. 2010; Bonafede et al. 2013; Kuchar and Enßlin 2011; Vacca et al.2012). Turbulent gas motions from AGN feedback have been observed directly with theHitomi satellite in the Perseus cluster (Aharonian et al. 2016) showing velocity dispersionsof ∼ 200 km/s on scales of < 60 kpc. This is not compatible with a medium based solelyon Coulomb collisions.

Thus it is unavoidable to consider a more complex prescription of the ICM plasma. In thefuture, stronger constraints on the velocity structure of gas motions in galaxy clusters willbe provided by the XIFU instrument on the Athena satellite (Ettori et al. 2013; Roncarelliet al. 2018).

We note that modern numerical simulations of galaxy clusters reach and exceed spatialresolutions of the Spitzer collisional mean free path. It follows that other processes thanCoulomb scattering have to maintain collisionality on smaller scales for these simulationsto be valid at all. Just adding a magnetic field to the Spitzer model, i.e. Coulomb scatteringplus a Lorentz force, does not suffice to make the ICM collisional on kpc scales. In a micro-physical sense the MHD magnetic field is a mean magnetic field that arises after averagingover micro-physical quantities (adiabatic invariants Schlickeiser 2002).


3.2.2 Turbulence and the Weakly-Collisional ICM

In the MHD limit, turbulence can excite three MHD waves, of which two have compressivenature (fast and slow modes, similar to sound waves) and one is solenoidal (Alfvén mode).The Alfvén speed is given by Alfvén (1942):

vA = B√4πρ

(9)

= 69B

1 μG

(nth

10−3 cm−3

)−1/2

km/s, (10)

with the number density of (thermal) ions nth.Numerical simulations of cluster formation find turbulent velocities at the outer scale of

several hundred km/s (Miniati 2014), which means that ICM turbulence starts off super-Alfvénic on the largest scales. Thus the magnetic field is dynamically not important near theouter scale and field topology is shaped by fluid motion.

Integrating Eq. (3) over k, we find that vl ∝ l1/3 and with Eq. (9) the Alfvén scale, wherethe magnetic field back-reacts on turbulent motions (Brunetti and Lazarian 2007):

lA ≈ 100

(B

μG

)3(L0

300 kpc

)(VL

103 km/s

)−3(nth

10−3 cm−3

) 32

pc, (11)

which is already smaller than the classical mean free path derived before and leads to aReynolds number of a few 1000. As we will see, this scale is crucial to numerically resolvemagnetic field growth by turbulence.

In principle, one has to consider three separate turbulent cascades, whose interplaychanges close around Alfvén scale (see Brunetti and Lazarian 2011b, and ref. therein).Here the character of turbulence dramatically changes. The Lorentz force introduces stronganisotropy to fluid motions, viscosity and turbulent eddies become anisotropic and non-localinteractions between modes in the turbulent cascade start to be important. See Goldreich andSridhar (1997), Schekochihin and Cowley (2007) for a more detailed picture of these pro-cesses.

That leaves us to ask, what is it that keeps the ICM collisional on scales much smallerthan the Alfvén scale, so MHD is applicable at all? Schekochihin et al. (2005b), Beresnyakand Lazarian (2006), Schekochihin and Cowley (2007), Schekochihin et al. (2008) proposethat due to the large Spitzer mean free path, the non-ideal MHD equations are not sufficientto estimate viscosity and obtain a Reynolds number for the ICM. Kinetic calculations revealthat particle motions perpendicular to the magnetic field are suppressed and motions parallelto the field can exist and excite firehose and mirror instabilities. The instabilities inject MHDwaves, which act as scattering agents (magnetic mirrors). Scattering off these self-exitedmodes isotropizes particle motions on very small length and time scales. This picture isconfirmed also by hybrid-kinetic simulations (Kunz et al. 2014).

Under these conditions, a lower limit of the viscous scale of the ICM is given by the mo-bility of thermal protons in a magnetic field, which is the Larmor radius (e.g. Schekochihinet al. 2005b; Beresnyak and Miniati 2016; Brunetti and Lazarian 2011b):

lmfp = rLarmor (12)

≈ 3 · 10−12 kpc

(T

10 keV

)(B

μG

)−1

. (13)


Fig. 2 Left: Cartoon illustrating the stretching and folding for magnetic field lines on small scales fromSchekochihin et al. (2002b). Right: Cartoon from Cho et al. (2009) depicting the growth of magnetic energyin driven turbulence simulations with very weak initial magnetic field. The initial seed field sets the timescalefor the end of the kinematic dynamo and the beginning of the non-linear dynamo

In this case, the effective Reynolds number of the ICM becomes:

Re,eff =(

l0

lν

)4/3

∼ 1019. (14)

This estimate predicts a highly turbulent ICM down to non-astrophysical scales and estab-lishes collisionality on scales of tens of thousands of kilometers. This is good news for sim-ulators, because the fluid approximation is well motivated in galaxy clusters and probablyvalid down to scales forever out of reach of simulations (Santos-Lima et al. 2014, 2017).

The bad news is that the physics of the medium is complicated, so that e.g. transportproperties of the ICM are dominated by scales out of reach for simulations and observa-tions. One example is heat conduction, where some estimates from kinetic theory predict noconduction in the weakly-collisional limit (Schekochihin et al. 2008; Kunz 2011). Indeed,only an upper limit was found by comparing observations with simulations (ZuHone et al.2015b). Thus, the properties of the medium cannot be constrained any further. Additionally,the likely presence of cosmic-ray protons makes the picture of generation and damping ofcompressive and Alfvén modes/turbulence even more involved (Fig. 5) (Schlickeiser 2002;Brunetti and Lazarian 2011c; Brunetti et al. 2004).

Now that we have established that MHD is very likely applicable down to sub-pc scales,we can discuss how (large-scale) magnetic fields can be amplified by turbulence in the MHDlimit.

3.3 The Small-Scale Dynamo

If a magnetic field is present in a turbulent flow, the properties of turbulence can changesignificantly due to the back-reaction of the field on the turbulent motions (Kraichnan andNagarajan 1967; Goldreich and Sridhar 1997). In a magnetic dynamo, the kinetic energy ofturbulence is transformed into magnetic energy, which is a non-trivial theoretical problem.The dissipation of magnetic energy into heat occurs at the resistive scale lη and the magneticReynolds number is defined as:

Rm = v0

k0η(15)

∝(

lη

l0

)4/3

(16)


The magnetic Prandtl number relates resistive with diffusive scales Eq. (14).

Pm = ν

η(17)

= Rm

Re

=(

lν

lη

)4/3

For a theoretical framework of the gyrokinetics on small scales, including a discussion oncluster turbulence, we refer the reader to Schekochihin et al. (2009).

In a simplified picture, magnetic field amplification by turbulence is a consequence of thestretching and folding of pre-existing field lines by the random velocity field of turbulence,which amplifies the field locally due to flux conservation (Fig. 2, left) (Batchelor 1950;Biermann and Schlüter 1951). If a flux tube of radius r1 and length l1 with magnetic fieldstrength B1 is stretched to length l2 and radius r2, mass conservation leads to:

r2

r1=

√l1

l2. (18)

The magnetic flux S1 = πr21 B is conserved in the high-β regime, so for an incompressible

fluid:

B2 = B1l2

l1(19)

By e.g. folding or shear (Fig. 2, left) the field can be efficiently amplified (Vaınshteın andZel’dovich 1972; Schekochihin et al. 2002a). Repeating this process leads to an exponen-tial increase in magnetic energy, if the field does not back-react on the fluid motion (Fig. 2,right). In a turbulent flow the folding occurs on a time scale of the smallest eddy turnovertime, i.e. close to the viscous scale. Flux tubes are tangled and merged, and their geome-try/curvature is set by the resistive and viscous scales of the flow. The energy available formagnetic field growth is the rate of strain δv/l (Schekochihin et al. 2005a). We note thatdue to the universality of scales in turbulence, the dynamo process does not depend on theactual magnetic field strength and time scale of the system. As long as the conditions for asmall scale dynamo are satisfied, field amplification will proceed as shown in Fig. 2, right.

For a small (10−13 G) initial seed field in galaxy environments or proto-clusters (seeSect. 2), back-reaction is negligible, Pm is very large and a small-scale dynamo (SSD) oper-ates in the kinematic regime of exponential amplification without back-reaction (Kulsrud andAnderson 1992). The SSD proceeds from small to large scales in an inverse cascade startingat the resistive scale. A rigorous treatment of this process based on Gaussian random fieldsin the absence of helicity was first presented by Kazantsev (1968), for an instructive applica-tion to proto-clusters see e.g. Federrath et al. (2011b), Schober et al. (2013), and Latif et al.(2013). For a unique experimental perspective on the kinematic dynamo see Meinecke et al.(2015). In Fig. 3, we reproduce the time evolution of magnetic energy (left) and of the mag-netic and kinetic power spectra (right) from an idealized simulation of the MHD dynamo(Cho et al. 2009). Here kν = 1/lnu ≈ 100, and the kinematic dynamo proceeds until t = 15.An instructive numerical presentation can be also found in Porter et al. (2015), a detailedexposure is presented in Schekochihin et al. (2004).

The exponential growth of the kinematic dynamo is stifled quickly (Brandenburg 2011),once the magnetic field starts to back-react on the turbulent flow. The dynamo then enters


Fig. 3 Left: Evolution of magnetic energy over time in a simulation of driven turbulence. The transition fromkinematic to non-linear dynamo occurs at t = 15. Right: Magnetic field energy spectra over wave number fordifferent times of the same run. Both figures by Cho et al. (2009)

the non-linear regime and turbulence grows a steep inverse cascade with an outer magneticscale lB. In Fig. 3, this occurs for t > 15 and kB = 1/lB ≈ 10 at t = 40. In principle, growthwill continue until equipartition with the turbulent kinetic energy is attained (Haugen andBrandenburg 2004; Brandenburg and Subramanian 2005; Cho et al. 2009; Porter et al. 2015;Beresnyak and Miniati 2016).

What does this mean for galaxy clusters? Above we had motivated a lower limit forthe viscous scale in proto-clusters of around 1000 km (Eq. (13)) and Reynolds numbers ofup to 1019. The resistive scale is highly uncertain, but likely small enough for an SSD tooccur. The large Reynolds number leads to a growth timescale of the kinematic dynamo ofτ ≈ 1000 yrs (Schekochihin et al. 2002b, 2004; Beresnyak and Miniati 2016). It is clear thatthis exponential growth is so fast that it will complete in large haloes before galaxy clustersstart forming at redshifts 2–1. The kinematic dynamo efficiently amplifies even smallest seedfields until back-reaction plays a role, i.e. the Alfvén scale approaches the viscous scale.

Depending on the physics of the seeding mechanism, the kinematic phase will take placein the environment of high redshift galaxies that is polluted by jets and outflows, in proto-clusters or, in the case of a cosmological seed field, in all collapsing over-dense environmentsat high redshift (Zeldovich et al. 1983; Kulsrud and Anderson 1992; Kulsrud et al. 1997;Latif et al. 2013).

However, contrary to the idealized turbulence simulations shown in Fig. 3, turbulentdriving in clusters occurs highly episodic and at multiple scales at once (Sect. 3.5), so theequipartition regime is never reached. Instead, the magnetic field strength and topology willdepend on the driving history of the gas parcel under consideration. It is also immediatelyclear that as opposed to amplification by isotropic compression, this dynamo erases all im-print of the initial seed field. Thus we cannot hope to constrain seeding processes frommagnetic fields in galaxy clusters, but instead have to look to filaments and voids, where thedynamo may not be driven efficiently.

3.4 Cosmic-Ray Driven Amplification and Plasma Effects

Magnetic fields can be amplified by a range of effects caused by cosmic rays. Current-driven instabilities, e.g. (Bell 2004), have been shown to amplify magnetic fields by con-siderable factors (Riquelme and Spitkovsky 2010). The electric current that drives this in-stability comes from the drift of CRs. The return electric current of the plasma leads to atransverse force that can amplify transverse perturbations in the magnetic field. Bell (2004)


Fig. 4 Ion number density (top) and magnetic field strength (bottom) for a parallel shock wave with Machnumber M = 20 at 1000 ω−1

c = mc/eB0 from (Caprioli and Spitkovsky 2014b)

pointed out that the fastest instability is caused by the return background plasma current thatcompensates the current produced by CRs streaming upstream of the shock. It is importantto note that this instability is non-resonant and can be treated using ideal MHD. The Bellor non-resonant streaming (NRS) instability has been tested in various numerical studiesusing a range of methods ranging from pure MHD (Zirakashvili and Ptuskin 2008), fullPIC (Riquelme and Spitkovsky 2011), hybrid (Caprioli and Spitkovsky 2014a,b, Fig. 4) toVlasov or PIC-MHD (Reville and Bell 2013; Reville et al. 2008; Bai et al. 2015); see Mar-cowith et al. (2016) for a review. In strong SNR shocks, a non-resonant long-wavelengthinstability can amplify magnetic fields as well (Bykov et al. 2009, 2011), but this has notbeen confirmed by simulations. A full non-linear calculation is needed to take into accountthe feedback of the CRs on the shock structure that may lead to a significant modificationof the shocks structure (e.g. Malkov and O’C Drury 2001; Vladimirov et al. 2006; Bykovet al. 2014). Recent γ -ray observations of SNR challenge this picture, so CR spectra mightbe steeper than the test-particle prediction (Caprioli 2012; Slane et al. 2014). All the afore-mentioned effects operate on length scales comparable to the gyro-radius of protons.

Filamentation instabilities can act on larger scales, as do models where CRs drive a tur-bulent dynamo (Drury and Downes 2012; Brüggen 2013). In the latter case, the turbulenceis caused by the cosmic-ray pressure gradient in the upstream region which exerts a force onthe upstream fluid that is not proportional to the gas density. Density fluctuations then lead tofluctuations in the acceleration which, in turn, produce further density fluctuations. CRs arealso able to generate strong magnetic fields at shock fronts which is invoked to explain thehigh magnetic field strengths in several historical supernova remnants. This was first stud-ied in the context of the high magnetic field strengths deduced from X-ray observations ofsupernova remnants. In fast shocks, the streaming of CRs into the upstream region triggersa class of plasma instabilities that can grow fast enough to produce very strong magneticfields (Lucek and Bell 2000).

More recently, Reville and Bell (2013) have studied a CR-driven filamentation instabil-ity that also results from CR streaming, but contrary to the Bell-instability generates long-wavelength perturbations. Caprioli and Spitkovsky (2014b) have investigated CR-driven fil-amentation instabilities using a di-hybrid method where electrons are treated as a fluid andprotons as kinetic particles. While progress in this field has grown substantially over the pastyears, very few PIC simulations for weak shocks in high-β plasmas have been done (e.g.Guo et al. 2016).

In analytical work (e.g. Melville et al. 2016), it has been shown that microphysical plasmainstabilities can produce a more efficient small-scale dynamo than its MHD counterpart de-


Fig. 5 Cartoon depicting thecascade of only compressiveturbulence over length scale ingalaxy clusters, consideringdamping from thermal ions andcosmic-ray protons (Donnert andBrunetti 2014)

scribed above. In this picture, shearing motions drive pressure anisotropies that excite mirroror firehose fluctuations (as seen in direct numerical simulations of collisionless dynamo; seeRincon et al. 2016). These fluctuations lead to anomalous particle scattering that lead to fieldgrowth. As shown in Mogavero and Schekochihin (2014), these scatterings can decrease theeffective viscosity of the plasma thereby allowing the turbulence to cascade down to smallerscales and thus develop greater rates of strain and amplify the field faster. Within a numberof large eddy turn-over times, this process can result in magnetic fields that saturate nearequipartition with the kinetic energy of the ICM.

While the total budget of cosmic ray protons stored in clusters is now constrained to≤ 1% (on average) for the thermal gas energy by the latest collection of Fermi-LAT data(Ackermann et al. 2014), it cannot be excluded that a larger fraction of cosmic rays mayexist close to shocks in the intra-cluster medium. At present, the limits that can be derivedfrom γ -rays are of ≤ 15%, at least in the case of the (nearby) relics in Coma (Zandanel andAndo 2014).

3.5 Processes that Drive Turbulence in Clusters

The accretion of gas and Dark Matter subunits is a main driver of turbulence in clusters.During infall, gas gets shock-heated around the virial radius (Mach numbers ∼ 10). In majormergers, the displacement of the ICM creates an eddy on the scale of the cluster core radii(e.g. Donnert and Brunetti 2014). Shear flows generated by in-falling substructure injectturbulence through Kelvin-Helmholtz (K-H) and Rayleigh-Taylor (R-T) instabilities (e.g.Subramanian et al. 2006; Su et al. 2017; Khatri and Gaspari 2016). Feedback from centralAGN activity, radio galaxies and galactic winds inject turbulence on even smaller scales(e.g. Churazov et al. 2004; Brüggen et al. 2005a; Gaspari et al. 2011). As a result of thiscomplex interplay of episodic driving motions on scales of half a Mpc to less than a kpc,the intra-cluster medium is expected to include weak-to-moderately-strong shocks (M ≤ 5)and hydrodynamic shear, leading to a turbulent cascade down to the dissipation scale.

The solenoidal component (Alfvén waves) of the cascade will drive a turbulent dynamo,while the compressive component (fast & slow modes) produces weak shocks and adiabaticcompression waves, which can in turn generate further small-scale solenoidal motions (e.g.,Porter et al. 2015; Vazza et al. 2017b). The relative contributions from both components


Fig. 6 Left: Gas motions in the first Eulerian simulation of a merging cluster (Schindler and Mueller 1993).Central and Right panel: projected enstrophy energy flux for a state-of-the-art Eulerian simulation with ENZOat z = 1 and z = 0, taken from Wittor et al. (2017b)

will depend on the turbulent forcing and its intensity (Federrath et al. 2011b; Porter et al.2015). Compressive and solenoidal components of the turbulent energy are also expectedto accelerate cosmic-ray protons and electrons via second-order Fermi processes, whichagain alters the properties of turbulence on small scales (see Brunetti and Jones 2014, for areview). In Fig. 5 we reproduce a cartoon plot of the compressive cascade in clusters fromDonnert and Brunetti (2014) that depicts the relevant scales: the classical mean free path(Eq. (6)), the Alfven scale (Eq. (11)) and the dissipation scales, if the cascade is damped bythermal protons (k > 10−2) or CR protons (k ≈ 1). The graph also includes the sound speedand the Alfvén speed and marks the regions accessible by current cosmological simulations.A more involved graph can be found in Brunetti and Jones (2014).

Note that the simple “Kolmogorov” picture of turbulence (Sect. 3.2) with a singlewell defined injection scale, an inertial range and a single dissipation scale is oversimpli-fied in galaxy clusters. As argued above, structure formation leads to an increase of theouter/driving scale with time and injection concurrently takes place at many smaller scalesand can be highly intermittent. Thus a strictly-defined inertial range does probably not existand turbulence may be more loosely defined in clusters than in other fields of astrophysics.Cosmological simulations can be used to capture the complexity of these processes.

4 Simulations of Turbulence and the Small Scale Dynamo in Clusters

4.1 Simulations of Cluster Turbulence

Simulations of merging clusters have been pioneered by Evrard (1990), Thomas and Couch-man (1992), who reported a shock traveling outward during a merger. Schindler and Mueller(1993) for the first time used a Eulerian PPM scheme with 603 zones to follow the gas dy-namics in an idealized merger (Fig. 6, left) (see also Roettiger et al. 1993, 1997). Usingidealized adaptive mesh refinement Eulerian merger simulations, Ricker and Sarazin (2001)for the first time report ram pressure stripping and turbulence, with eddy sizes of “severalhundred kpc” [..] “pumped by DM driven oscillations of the gravitational potential”. Tak-izawa (2005), Asai et al. (2004) used a TVD scheme to study the driving of shocks andturbulence by substructure in idealized cluster simulations. They focused on the injection ofinstabilities and gas stripping (see Sect. 6).


In cosmological simulations, turbulence was first studied by Dolag et al. (2005b) usingSPH with a low viscosity scheme (for shocks see Miniati et al. 2000). They find subsonicvelocity dispersions of 400–800 km/s on scales of 20 to 140 kpc, with turbulent energyfractions of 5–30 per cent and a trend for higher turbulent energies in higher mass clusters.Turbulent energy spectra from their simulations were flatter than the Kolmogorov expecta-tion, but might have been limited by numerics (see Sect. 4.4). Their work was extended toa sample of 21 clusters by Vazza et al. (2006), who provided scaling laws for the turbulentenergy over cluster mass, see Valdarnini (2011) for a later study.

In a seminal contribution, Ryu et al. (2008) studied the generation and evolution of turbu-lence in a Eulerian cosmological cluster simulation. They showed that turbulence is largelysolenoidal, not compressive, with subsonic velocities in clusters and trans-sonic velocities infilaments. In agreement with prior SPH simulations, they find a clear trend of rms velocitydispersion with cluster mass and turbulent energy fractions/pressures of 10–30%. They alsopropose a vorticity based dynamo model, which we will discuss in Sect. 4.2.

The influence of turbulent pressure support on cluster scaling relations was studied byNagai et al. (2007), Lau et al. (2009), Shaw et al. (2010), Burns et al. (2010), Battaglia et al.(2012), Nelson et al. (2014), Schmidt et al. (2017). Consistently, turbulent pressure increaseswith radius in simulated clusters, which is related to the increased thermal pressure causedby the central potential of the main DM halo. An analytic model for non-thermal pressuresupport was presented by Shi and Komatsu (2014), and also validated by numerical simu-lations (Shi et al. 2015, 2016). First power spectra of turbulence in Eulerian cosmologicalcluster simulations were presented by Xu et al. (2009), Vazza et al. (2009). Their kineticspectra roughly follow the Kolmogorov scaling. The simulations reach an “injection region”of turbulence larger than 100 kpc, an inertial range between 100 kpc and 10 kpc and a dissi-pation scale below 10 kpc. Thus their Reynolds number was 10–100.

The next years saw improvements in resolution of cluster simulations, due to the in-evitable growth in computing power. Increasingly higher Reynolds numbers could bereached and/or additional physics could be implemented usually with adaptive mesh re-finement (AMR). Vazza et al. (2011) studied a sample of simulated clusters with Reynoldsnumber of up to 1000. They also developed new filtering techniques to estimate turbulentenergy locally. They showed that the turbulent energy in relaxed clusters reach only a fewpercent. Maier et al. (2009), Iapichino et al. (2011) added a subgrid-scale model for unre-solved turbulence to their simulations and studied the evolution of turbulent energy. Theyfound that peak turbulent energies are reached at the formation redshift of the underlyinghalo. Their subgrid model shows that unresolved pressure support is usually not a problemin cluster simulations, and that half of the simulated ICM shows large vorticity. Paul et al.(2011) simulated a sample of merging clusters and found a scaling of turbulent energy withcluster mass as ∝ M5/3, consistent with earlier SPH results (Vazza et al. 2006). The influ-ence of minor mergers on the injection of turbulence in a idealized scenario of a cool corecluster was simulated with anisotropic thermal conduction by Ruszkowski and Oh (2011).They found that long-term galaxy motions excite subsonic turbulence with velocities of100–200 km/s and give a detailed theoretical model for the connection between vorticityand magnetic fields.

Vazza et al. (2012) used an improved local filter to estimate the turbulent diffusivityin their simulations as Dturb ≈ 1029–30 cm2/s and identify accretion and major mergers asdominant drivers of cluster turbulence.

In a series of papers, Miniati (2014, 2015) introduced static Eulerian mesh refinementsimulations to the field. They reach a peak resolution of ≈ 10 kpc covering the entire virialradius of a massive galaxy cluster with a PPM method. Consistent with previous studies


Fig. 7 Vorticity map for the innermost regions of a simulated ∼ 1015 M� galaxy cluster at high resolution,in the “Matrioska run” by Miniati (2014)

they find that shocks generate 60% of the vorticity in clusters. Their adiabatic simulationsshow turbulent velocity dispersions above 700 km/s, regardless of merger state. The anal-ysis using structure functions reveals that solenoidal/incompressible turbulence with a Kol-mogorov spectrum dominates the cluster, while compressive turbulence with a Burgers slope(Burgers 1939) become more important towards the outskirts. They propose that a hierar-chy of energy components exists in clusters, where gravitational energy is mostly dissipatedinto thermal energy, then turbulent energy and finally magnetic energy with a constant effi-ciency (Miniati and Beresnyak 2015). Vorticity maps from their approach are reproduced inFig. 7.

In the most recent studies, the resolution has been improved to simulate the first earlybaroclinic injection of vorticity in cluster outskirts (e.g. Vazza et al. 2017b; Iapichino et al.2017) as well as its later amplification via compression/stretching during mergers (Wittoret al. 2017a). Using the Hodge-Helmholtz decomposition, high resolution Eulerian simula-tions measure a very large fraction of turbulence being dissipated into solenoidal motions(Miniati 2014; Vazza et al. 2017b; Wittor et al. 2017a). Baroclinic motions inject enstrophyon large scales, while dissipation and stretching terms govern its evolution.

Recent simulations using Lagrangian methods focus on including more subgrid physicsin the setup to study the influence of magnetic fields on galaxy formation. Marinacci et al.(2015) show that the redshift evolution of the rms velocity fluctuations in the “Illustris TNG”galaxy formation simulations is independent of seed magnetic fields.

4.2 Cosmological Simulations of Magnetic Fields in Galaxy Clusters

Pioneering studies of magnetic fields in simulated large-scale structures were conducted byDe Young (1992), Kulsrud et al. (1997), Roettiger et al. (1999). First full MHD simulationsof cluster magnetic fields from nG cosmological seeds have been presented by Dolag et al.(1999, 2002), Bonafede et al. (2011). They found a correlation of the magnetic field strengththe ICM gas density with an exponent of 0.9, using smooth particle hydrodynamics (SPH)(Dolag and Stasyszyn 2009; Beck et al. 2016). This is close to the theoretical expectationfor spherical collapse (Fig. 1, Eq. (2)) and it is in-line with observations from Faraday ro-tation measures. In the center of clusters, their simulations obtain a magnetic field strength


Fig. 8 Projection of magnetic field strength in three cosmological simulations using different MHD ap-proaches and solvers. Left: non-radiative GADGET SPH simulations with galactic seeding by Donnert et al.(2009), based on the MHD method by Dolag and Stasyszyn (2009). Middle: non-radiative ENZO MHD simu-lation on a fixed grid by Vazza et al. (2014) using the Dender cleaning (Dedner et al. 2002); Right: Simulationwith full “Illustris TNG” galaxy formation model using a Lagrangian finite volume method (Marinacci et al.2018b)

Fig. 9 3-dimensional kinetic and magnetic power spectra in ENZO MHD simulations by Xu et al. (2009)(assuming a seeding of magnetic fields by AGN) and by Vazza et al. (2018), assuming a primordial magneticfield of 0.1 nG (comoving), as a function of resolution

of 3–6 μG, over a wide range of cluster masses. Subsequently the simulations were usedto model giant radio haloes (Dolag and Enßlin 2000; Donnert et al. 2010), the influence ofthe field on cluster mass estimates (Dolag and Schindler 2000; Dolag et al. 2001), the prop-agation of ultra high energy cosmic-rays (Dolag et al. 2005a) and the distribution of fastradio bursts (Dolag et al. 2015). Donnert et al. (2009), Beck et al. (2013a) presented modelsfor cluster magnetic fields seeded by galaxy feedback, and established that different seedingmodels can lead to the same cluster magnetic field. Beck et al. (2012) showed theoreticaland numerical models for magnetic field seeding and amplification in galactic haloes. Wereproduce projected magnetic field strengths in cosmological simulations from three differ-ent methods, GADGET (SPH), ENZO (Eulerian finite volume) and AREPO (Lagrangian finitevolume) in Fig. 8.

Ryu et al. (2008) established the connection between shock driven vorticity duringmerger events and magnetic field amplification in clusters using Eulerian cosmological sim-


Fig. 10 Phase diagrams for different cosmological simulations, like Fig. 1. Top left: RAMSES CT simula-tion of a cooling-flow galaxy cluster (Dubois and Teyssier 2008; Dubois et al. 2009). Top right: ENZO CTsimulation of a major merger cluster (fields injected by AGN activity) (Skillman et al. 2013). Bottom left,right: AREPO (Powell scheme) simulation without and with Illustris galaxy formation model, respectively(Marinacci et al. 2015)

ulations. They applied a semi-analytic model of the small scale dynamo coupled to theturbulent energy to derive μG fields in clusters (see also Beresnyak and Miniati 2016).

Xu et al. (2009, 2011) used AGN seeding in the first direct Eulerian MHD cluster sim-ulations to obtain magnetic field strengths of 1–2 μG in clusters with a second order TVDmethod and constrained transport (Li et al. 2008). We reproduce power spectra from thissimulation in Fig. 9, left. Considering cosmological seed fields, Vazza et al. (2014) usedlarge uniform grids to simulate magnetic field amplification in a massive cluster. Ruszkowskiet al. (2011) presented a simulation of cluster magnetic fields with anisotropic thermal con-duction. They find that conduction eliminates the radial bias in turbulent velocity and mag-netic fields that they observe without conduction.

Within the limit of available numerical approaches, modern simulations find that adia-batic compression/rarefaction of magnetic field lines is the dominant mechanism across mostof the cosmic volume (see Fig. 10), with increasing departures at high density, ρ ≥ 102〈ρ〉,when dynamo amplification sets in. Additional scatter in this relation is also found in pres-ence of additional sources of magnetization or dynamo amplification, such as e.g. feedbackfrom AGN, as shown by the comparison between non-radiative and “full physics” runs. Us-ing a Lagrangian finite volume method, Marinacci et al. (2015, 2018a,b) showed magnetic


Fig. 11 Map of projected mean magnetic field strength for re-simulations of a cluster with increasing reso-lution, for regions of 8.1 × 8.1 Mpc2 around the cluster center at z = 0. Each panel shows the mass-weightedmagnetic field strength (in units of log10[μG] for a slice of ≈ 250 kpc along the line of sight). Adapted fromVazza et al. (2018)

field seeding and evolution with the “Illustris” subgrid model for galaxy formation, also in-cluding explicit diffusivity. They obtained μG magnetic fields in clusters when they includedseeding from galaxy feedback (Fig. 10, bottom).

Recently, Vazza et al. (2018) simulated the growth of magnetic field as low as 0.03 nGup to ∼ 1–2 μG using AMR with a piece-wise linear finite volume method. By increasingthe maximum spatial resolution in a simulated ∼ 1015M� cluster, they observed the onset ofsignificant small-scale dynamo for resolutions ≤ 16 kpc, with near-equipartition magneticfields on ≤ 100 kpc scales for the best resolved run (≈ 4 kpc), see Fig. 11. They estimatedthat ∼ 4% turbulent kinetic energy was converted into magnetic energy. The amplified 3Dfields show clear spectral, topological and dynamical signatures of the small-scale dynamoin action, with mock Faraday Rotation roughly in-line with observations of the Coma cluster(Bonafede et al. 2013). A significant non-Gaussian distribution of field components is con-sistently found in the final cluster, resulting from the superposition of different amplificationpatches mixing in the ICM.

4.3 Cosmological Simulations of Magnetic Fields Outside of Galaxy Clusters

The peripheral regions of simulated galaxy clusters mark the abrupt transition from su-personic to subsonic accretion flows, and the onset of the virialization process of the in-falling gas. The accreted gas moves supersonically with respect to the warm-hot inter-galactic medium in the cluster periphery, which triggers M ∼ 10–100 strong shocks inthe outer regions of clusters and in the filaments attached to them (e.g. Ryu et al. 2003;


Pfrommer et al. 2006). Downstream of such strong shocks, supersonic turbulence is in-jected towards structures, together with a first inject of vorticity by oblique shocks (e.g.Kang et al. 2007; Ryu et al. 2008; Wittor et al. 2017b). In these physical conditions,the SSD is predicted to be less efficient, because of the predominance of compressiveforcing of turbulent motions (Ryu et al. 2008; Federrath et al. 2011a; Jones et al. 2011;Schleicher et al. 2013; Porter et al. 2015). In this case, the maximum magnetic field aris-ing from SSD amplification in the 105 K ≤ T ≤ 107 K medium of filaments would be∼ 0.01–0.1 μG (e.g. Ryu et al. 2008; Vazza et al. 2014). Direct numerical simulations in-vestigated the small-scale dynamo amplification of primordial fields in cosmic filaments,so far reporting no evidence for dynamo amplification, unlike for galaxy clusters simu-lated with the same method and at a similar level of spatial detail (Vazza et al. 2014). Thistrend is explained by the observed predominance of compressive turbulence at all reso-lutions (unlike in clusters, where turbulence gets increasingly solenoidal as resolution isincreased), as well as by the limited amount of turnover times that infalling gas expe-riences before being accreted onto clusters (e.g. Ryu et al. 2008; Vazza et al. 2014). Ifthese results will be confirmed by simulations with even larger resolutions, it has the im-portant implication that the present-day magnetization of filaments should be anchored tothe seeding events of cosmic magnetic fields, posing a strong case for future radio obser-vations (e.g. Gheller et al. 2016; Vazza et al. 2017a). In this scenario the outer regionsof galaxy clusters and filaments are expected to retain information also on the topologyof initial seed fields even today, as shown in numerical simulations at high resolution(e.g. Brüggen et al. 2005b; Marinacci et al. 2015, see also Fig. 12), in case the mag-netic fields have a primordial origin. Conversely, if the fields we observe in galaxy clus-ters are mostly the result of seeding from active galactic nuclei and galactic activities, themagnetization at the scale of filaments and cluster outskirts is predicted to be low (e.g.Donnert et al. 2009; Xu et al. 2009; Marinacci et al. 2015). Future surveys in polariza-tion should have the sensitivity to investigate the outer regions of galaxy clusters down to∼ 1–10 rad/m2 (e.g. Taylor et al. 2015; Bonafede et al. 2015; Vacca et al. 2016), whichis enough to discriminate among most extreme alternatives in cluster outskirts (e.g. Vazzaet al. 2017a).

4.4 Discussion

Simulations of magnetic field amplification in clusters have reproduced key observations fortwo decades now. Most of the early progress has been achieved with Lagrangian methodsoriginally developed in the galaxy formation context, most notably SPH (Dolag et al. 2002).These simulations reproduce the magnetic field strength inferred from rotation measures inclusters and have been used extensively to model related astrophysical questions. However,the adaptivity of Lagrangian methods and the particle noise in SPH limits their ability toresolve the structure of the magnetic field, especially in low density environments (clusteroutskirts, filaments).

Clear theoretical expectations for the small scale dynamo in clusters have been estab-lished (Ryu et al. 2008; Beresnyak and Miniati 2016), also from idealized simulations (e.g.Schekochihin et al. 2004; Cho et al. 2009; Porter et al. 2015). Some of these expectationshave been tested in cosmological simulations using Eulerian codes (e.g. Vazza et al. 2018).Recent Eulerian simulations approach observed field strengths in clusters, but do not reachfield strengths obtained from Lagrangian approaches. Beresnyak and Miniati (2016) arguedthat due to numerical diffusion, Eulerian approaches spend too much time in the exponen-tial/kinetic growth phase, thus the non-linear growth phase is severely truncated. Following


Fig. 12 Left: magnetic field vectors for a cosmic filaments simulated with AMR using FLASH (Brüggenet al. 2005b). Right: magnetic field vectors around a massive galaxy cluster (top) and a filament (bottom)simulated with AREPO (Marinacci et al. 2015) and for two different topologies of uniform seed magneticfields

Schekochihin et al. (2004), a clear indicator for the presence of a dynamo that cannot be pro-duced via compression is the anti-correlation of magnetic field strength and its curvature K:

K = (B · ∇)BB2

, (20)

so that BK12 = const, where the exponent has to be obtained from the magnetic field dis-

tribution. In cluster simulations, only Vazza et al. (2018) have demonstrated consistent cur-vature correlations. We note that in galactic dynamos, consistent results have recently beenachieved with Eulerian and Lagrangian codes (Butsky et al. 2017; Rieder and Teyssier 2016;Pakmor et al. 2017; Steinwandel et al. 2018), but only Steinwandel et al. (2018) showed acurvature relation.

In clusters, all simulations show an exponential increase in magnetic field strength fol-lowed by a non-linear growth phase (e.g. Beck et al. 2012). However, the timescale of ex-ponential growth is set by the velocity power/rate of strain at the resolution scale, which inturn is determined by the MHD algorithm (resolution, dissipation/noise). The real kinematicdynamo in primordial haloes is far below the resolution scale of every numerical scheme(Beresnyak and Miniati 2016) and needs to be treated with an large eddy approach (Yakhotand Sreenivasan 2005; Cho et al. 2009).

As we have motivated above, dynamo theory predicts that the final structure of clustermagnetic fields is shaped by turbulence near the Alfvén scale, because this is where the eddyturnover time is smallest (Eq. (11), a few kpc in a massive cluster merger). Thus an accu-rate simulation of field topology has to faithfully follow the velocity field and the magneticfield near this scale in the non-linear growth phase, i.e. at least achieve Reynolds numbers(Eq. (5)) of 300–500 at redshifts z < 1 during a major merger (Haugen et al. 2004; Beres-nyak and Miniati 2016). For an outer scale of 300 kpc, this implies evolution of turbulencevelocity and magnetic field growth at about 1 kpc, including numeric effects.

This makes the small scale dynamo in clusters is a very hard problem, because it com-bines the large dynamical range of scales in cosmological clustering with the evolution of


two coupled vector fields (turbulence and magnetic fields) near the resolution scale. Addi-tionally, seeding on smaller scales by galactic outflows may play an important role. Hence,it is likely the numerical dissipation scale that shapes the outcome of MHD simulations ina cosmological context. We now provide a short discussion of effective Reynolds numbersand numerical limitations in current approaches.

4.4.1 Effective Reynolds Numbers

From the numerical viewpoint, the Reynolds number of a flow increases with the effectivedynamic range reached inside a given volume. Its upper limit is set by the driving scale andthe spatial resolution in the volume of interest following Eq. (5). However, in any numericalscheme the effective dynamic range and Reynolds number of the flow are reduced by the cut-off of velocity and magnetic field power near the numerical dissipation scale in Fourier space(e.g. Dobler et al. 2003). Simply put, numerical error takes away velocity and magnetic fieldpower close the resolution scale in most schemes. The shape of the velocity power spectrumon small scales determines how much velocity power (rate of strain δu/l, see Sect. 3.3) isavailable to fold the magnetic field and drive the small-scale dynamo. Thus a less diffusive(finite volume) code reaches higher effective Reynolds numbers, faster amplification and amore tangled field structure at the same resolution.

We can quantify this behavior by introducing an effective Reynolds number of an MHDsimulation of turbulence as:

Re,min ≈(

L

ε�x

)4/3

, (21)

where �x is the resolution element, ε is a factor depending on the diffusivity of the numer-ical method, and L is the outer scale (in clusters 300–500 kpc, Sect. 3.5). As a conservativeestimate, one may assume in modern SPH codes ε ≥ 10 (Price 2012a, Fig. 13), in hybridcodes ε ≈ 10 (Hopkins (2015)). For second order finite difference/volume codes one oftenassumes ε ≈ 7 (e.g. Kritsuk et al. 2011; Rieder and Teyssier 2016). In Fig. 13 left, we repro-duce velocity power spectra from a driven compressible turbulence in a box simulation with1283 zones using the finite volume (FV) code AREPO and the discontinuous Galerkin (DG)code TENET (Bauer et al. 2016). Second order FV is shown in yellow, while second, thirdand fourth order DG power spectra are shown in green, blue and purple, respectively. Theformal resolution/Nyquist scale remains constant in all runs. However, with increasing orderof spatial and time interpolation, viscosity reduces, the effective dissipation scale shrinks,velocity power on small scales increases, the inertial range grows in size, and with it theeffective Reynolds number of the simulation (i.e. ε decreases). Note that the DG scheme hasmore power near the dissipation scale than the FV scheme, even at the same order (green vs.yellow). This indicates that formal convergence order is not sufficient to determine effectiveReynolds numbers at a given resolution. ε obviously depends on implementation details andhas to be determined empirically with driven turbulence “in a box” simulations. For a recentreview on high-order finite-volume schemes, see Balsara (2017).

4.4.2 Dynamos in Eulerian Schemes

In non-adaptive Eulerian cluster simulations the effective Reynolds number is set by theresolution of the grid and the diffusivity of the numerical method (e.g. Kritsuk et al. 2011).Federrath et al. (2011b) and Latif et al. (2013) reported that only by resolving the Jeans


Fig. 13 Left: Velocity power spectra of driven compressible turbulence with a second order finite volumescheme (yellow) and Discontinuous Galerkin schemes (2nd order: green, 3rd order blue, 4th order purple)(Bauer et al. 2016). Right: Velocity power spectra of decaying turbulence simulated with modern SPH (Becket al. 2016). The spectra were obtained using wavelet kernel binning to remove aliasing above the kernel scalekhsml

length of a halo with ≥ 64 cells the small-scale dynamo can develop (e.g. RM ∼ 32 settingε = 2 in Eq. (21)) in a proto-galactic halo of 106M� at z ∼ 10. However, Vazza et al. (2014)reported that small-scale amplification can begin before z = 0 in ∼ 1014M� galaxy clustersif their virial diameter is resolved with at least ≥ 100 cells (RM ∼ 50), while in order toapproach energy equipartition between turbulence and magnetic fields by z = 0 one needsto resolve the virial diameter with ≥ 1500 elements (RM ∼ 750 in the ideal case). Thesedifferences likely arise from the shapes of the numerical dissipative and resistive scales.The underlying Eulerian methods were either second or first order accurate and used CT orDedner cleaning to constrain magnetic field divergence.

Eulerian structure formation simulations produce flows with supersonic velocities rela-tive to the simulation grid. At the same time, the truncation error of Eulerian methods isinherently velocity dependent (Robertson et al. 2010; Bauer et al. 2016). It has been shownthat these errors do not pose a problem for the simulation of clusters in a cosmological con-text (Mitchell et al. 2009), but they may suppress the growth of instabilities close to thedissipation scale (e.g. Springel 2010) and thus further reduce the effective Reynolds numberof the simulation. We note that poorly un-split Eulerian schemes may also affect angularmomentum conservation close to the resolution scale and further reduce the accuracy ofe.g. galaxy formation simulations, where angular momentum conservation is desirable toproduce disc galaxies.

These arguments extend also to magnetic fields, whose advection poses a challenging testfor all Eulerian schemes. In Fig. 14, right, we reproduce the time evolution of magnetic en-ergy during the advection of a magnetic field loop in 2D with the ATHENA code at differentresolutions (Gardiner and Stone 2008). As the size of the field loop approaches the resolu-tion scale, field energy is diffused more quickly. Again, the diffusivity added by the schemeto keep local magnetic field divergence small varies with implementation and has to be de-termined by empirical tests. There are sizable differences even among CT schemes, whichinherently conserve the divergence constraint to machine precision (see e.g. Lee 2013).

4.4.3 Dynamos in Lagrangian Schemes

In adaptive Lagrangian cluster simulations, the resolution is a function of density and thusvaries in space and time during the formation of a cluster or filament. Thus the dissipation


Fig. 14 Left: Growth of magnetic energy in supersonic driven turbulence simulations in SPMHD (dashedlines) and FV (solid lines) at different resolutions (Tricco et al. 2016). Right: Evolution of magnetic energyin the advection of a magnetic field loop in 2 dimensions from Gardiner and Stone (2008)

scale and the Reynolds number are not well defined in Fourier space and turbulence canbe strictly defined only on the coarsest resolution element in a given volume. The effectof the adaptivity on the dynamo and especially the resulting field structure is not entirelyclear. It seems reasonable to assume additional (magnetic) dissipation, if a magnetized gasparcel moves to a less-dense environment and is adiabatically expanded and divergencecleaned. While turbulent driving is correlated with over-densities in a cosmological context,the turbulent cascade is not. Thus density adaptivity, which is a very powerful approach ingalaxy formation simulations, might introduce a density bias to the magnetic field distribu-tion in strongly stratified media. The growth rate of the turbulent dynamo depends on theeddy turnover time, which is smallest in highly resolved regions. Thus Lagrangian schemesmight grow magnetic fields faster in high density regions (cluster cores) than in low densityregions (cluster outskirts). However, it remains unclear how strongly current results are af-fected by this issue, simply because no Eulerian simulation with kpc resolution in the clusteroutskirts is available.

In cosmological simulations, Dolag et al. (1999, 2002) reported sizeable cluster magneticfields even with a traditional SPH algorithm and comparably low resolution. As we haveshown, theory provides clear predictions for the evolution of a magnetic field in a turbulentdynamo, which have been successfully verified with Eulerian methods. For some Lagrangianmethods (e.g. Pakmor et al. 2011), it is reasonable to assume that at fixed resolution theresult will be similar to the established dynamo theory, simply because their dissipationscale defaults to a finite volume method. For other new hybrid methods (Hopkins and Raives2016) the situation is less clear. In general, the idealized magnetic dynamo in Lagrangianschemes is not well researched yet and we would encourage the community to close thisgap.

For traditional SPH algorithms, its ability to accurately model hydrodynamic turbulencewas heavily debated (Bauer and Springel 2012; Price 2012a). We note that computing agrid representation from an irregularly sampled vector field to obtain a power spectrum is adiffusive process and prone to aliasing (Beck et al. 2016). Modern SPH schemes have im-proved significantly, and it has been shown that sub-kernel re-meshing motions are requiredto maintain sampling accuracy (Price 2012b). The influence of these motions on the mag-netic dynamo are not well understood, especially in the subsonic regime that is dominant inclusters.


In the supersonic regime, Tricco et al. (2016) compared simulations with M = 10 usingthe SPMHD code PHANTOM and the finite volume code FLASH with an HLL3R solver(Waagan et al. 2011) and both with Dedner cleaning. They found that the growth of magneticenergy in the SPMHD dynamo speeds up with increasing resolution. In contrast, the finitevolume scheme converged (Fig. 14, left). They found Prandtl numbers of Pr = 2 and Pr < 1,respectively. They argued that the growth in the SPMHD dynamo is due to the artificialviscosity and resistivity employed, which is negligibly small in the absence of shocks.

We note that these results cannot be simply transferred to galaxy cluster simulations. Asmentioned before, cluster turbulence is largely sub-sonic, super-Alfvénic and solenoidal,thus shocks do not play a role for the dissipation of turbulent energy. Cosmological codesusually do not include explicit dissipation terms, in contrast diffusivity is usually minimized.Driven subsonic turbulence simulations with SPMHD are required to characterize the sub-sonic SPMHD dynamo in clusters and clarify the role particle noise could play even in earlySPMHD cluster simulations. We note that some numerical amplification has been reportedin SPMHD simulations of the galactic dynamo (Stasyszyn and Elstner 2015; Dobbs et al.2016).

5 Magnetic Field Amplification at Shocks

Shocks amplify magnetic fields by a number of mechanisms, not all of which are well under-stood (Brüggen et al. 2012). Compression at the shock interface leads to the amplificationof the quasi-perpendicular part of the upstream magnetic field. Compressional amplifica-tion has the allure of explaining the large degrees of polarization in radio relics, but suffersfrom the limitation of small amplification factors. For amplification by pure compression,(Iapichino and Brüggen 2012) find for the ratio of magnetic fields:

Bdw

Buw=

√2σ 2 + 1

3, (22)

with the shock compression ratio σ . Thus, for typical shock strengths in cluster mergers,(M ≈ 2–3), the amplification factor is limited to around 2.5, which results in inconsistenciesof the minimum magnetic field strengths inferred in some radio relics with global magneticfield scalings (Donnert et al. 2017). Similar expressions have been found for SNR (Reynolds1998).

5.1 Shock-Driven Dynamo

Downstream of shocks, magnetic fields can be amplified by a small-scale dynamo that isdriven by turbulence created at the shock front (Binney 1974). This has been observed insupernova remnants (SNR) (Parizot et al. 2006). This turbulence could be driven by the baro-clinic vorticity that is generated for example by upstream inhomogeneities in gas density. Forparameters relevant in SNR, Giacalone and Jokipii (2007) have demonstrated in MHD sim-ulations that density inhomogeneities in the pre-shock fluid cause turbulence and magneticfield amplification in the post-shock fluid. Simulations by Inoue et al. (2009) showed thatthe maximum amplification is set by the plasma beta parameter. Sano et al. (2012) arguedthat turbulence is injected by Richtmyer-Meshkow instabilities. Fraschetti (2013) derived ananalytical approach for 2D SNR shocks. Guo et al. (2012) studied the interaction of a SNRshock propagating into a turbulent medium upstream. However, the relevant parameters in


Fig. 15 Left: Magnetic field amplification over Alfvénic Mach number in 2D MHD shock simulations fromJi et al. (2016). Right: Models for magnetic field evolution inferred in the Sausage relic from Donnert et al.(2016)

the shock and the upstream medium in SNR blast waves differ significantly from galaxycluster shocks. In clusters, Mach numbers are lower (< 5) and the plasma beta parameter islarger (βpl ≥ 100). It is unclear if there results from SNR carry over to the ICM.

Literature on turbulent magnetic field amplification in ICM shocks remains scarce.Iapichino and Brüggen (2012) studied the evolution of vorticity behind the shock. They ar-gue that self-generated vorticity from the shock is not sufficient to drive a turbulent dynamodownstream, but that about 30% of turbulent pressure is required upstream of the shock toexplain observed magnetic field lower limits.

Ji et al. (2016) studied magnetic field amplification in idealized MHD simulations ofshocks. They found that amplification is independent of plasma beta for Mach numbers of afew, but is linearly dependent on the Alfvénic Mach number in shocks. In Fig. 15, left, weshow their results for 2D simulations at different resolutions, with the highest resolution inmagenta. Below MA ≈ 10, compression dominates the amplification and results in magneticfield structures perpendicular to the shock normal. Above MA ≈ 10, turbulence injected bythe shock amplifies magnetic fields to strengths significantly higher than expected by com-pression. In this limit, the field topology becomes mostly quasi-parallel, because velocityshear is largest in the direction of shock propagation.

Along these lines, in Wittor et al. (2017a), it has been found that the stretching motionsdominate the evolution of turbulence in galaxy clusters. However, baroclinic motions areneeded to generate turbulence. The enstrophy dissipation rate peaks when the enstrophy ismaximal and this is the time when magnetic field amplification by a small-scale dynamowould be the strongest.

These results have important implications for radio relics. In most relics the lowerlimit for the downstream magnetic field is found to be around 1–3 μG (e.g. Finoguenovet al. 2010). This is consistent with equipartition magnetic field strengths of 4–7 μG (e.g.Nakazawa et al. 2009; Finoguenov et al. 2010; Stroe et al. 2014).

The ordered topology of magnetic fields expected by compressional amplification canexplain the large degree of polarization found in some radio relics, thus dis-favouring tur-bulent amplification. However, given typical Mach numbers (2–3), the lower limits on mag-netic field strengths in relics imply upstream fields of about 1–2 μG ahead of some shocks(Eq. (22)). As relics reside in the outskirts of clusters, this is difficult to explain with thecommon scaling of magnetic field strength with density/radius in the ICM (Bonafede et al.2010).


Fig. 16 Amplified magnetic fields produced by a remnant-core cold front in an MHD simulation detailed inAsai et al. (2007). The panels show slices of temperature (left), temperature gradient (middle), and magneticfield strength (right). An amplified magnetic field appears in a draping layer around the cold front

However, a recent model (Donnert et al. 2016) of the Sausage relic motivates AlfvénicMach numbers of around 100 in the shock and showed that exponential downstream fieldamplification (Fig. 15, right) can explain the steepening in the radio spectrum above 8 GHzfound in the Sausage (Stroe et al. 2013). More discussion will also be found in van Weerenet al. (this volume).

6 Magnetic Field Amplification from Cold Front Motions

Aside from turbulence and shocks, many galaxy clusters also possess subsonic bulk flowswhich can amplify magnetic fields in localized regions. The first evidence of these mo-tions was provided shortly after the launch of the Chandra X-ray Observatory. Chandra’ssub-arcsecond spatial resolution revealed the presence of surface brightness edges in manyclusters. Through spectroscopic analysis most of these edges, which superficially appearas shocks, were identified to be contact discontinuities, where the denser (brighter) side ofthe edge is colder than the lighter (dimmer) side. These features have been dubbed “coldfronts”, and are believed to be the result of subsonic gas motions driven by cluster mergersand cosmic accretion (for recent reviews see Markevitch and Vikhlinin 2007; Zuhone andRoediger 2016). Cold fronts have been described as forming via at least three processes:“remnant-core” fronts are formed by cool cores of sub-clusters or galaxies falling into ormerging with larger, more diffuse structures, “sloshing” cold fronts which are formed incool-core clusters by the displacement of the central low-entropy gas of the DM-dominatedcore, and “stream” cold fronts which are formed by collisions between coherent streams ofgas (Birnboim et al. 2010; Zuhone and Roediger 2016; Zinger et al. 2018).

The relevance of such bulk motions for the amplification of the cluster magnetic field wasfirst shown by Lyutikov (2006). They demonstrated that the subsonic motion of a dense gascloud through the ICM would amplify and stretch magnetic fields, regardless of the initialgeometry, along the contact discontinuity that forms, producing a thin “magnetic drapinglayer”. The only condition is that the Alfvénic Mach number MA > 1, a condition readilysatisfied in the ICM. The width of the layers is given roughly by �r ∼ L/M 2

A, For typicalconditions in the ICM and a mildly subsonic cloud with M � 1.0, �r ∼ 0.01L. Lyutikov(2006) also pointed out that such layers should be associated with a depletion of plasma. Thisis so because the total pressure should remain continuous within and around such a layer,given the subsonic motion of the gas, and thus an increase in magnetic pressure requires adecrease in thermal pressure. This indicates that such layers may be visible in X-ray obser-vations of cold fronts, though in practice there will be large uncertainties given projectioneffects.


Fig. 17 Amplified magnetic fields produced by sloshing cold fronts in an MHD simulation. Left panel: Slicethrough the gas temperature in keV, showing the spiral-shaped cold fronts. Right panel: Slice through themagnetic field strength, showing amplified fields within the cold fronts. Figure reproduced from Zuhone andRoediger (2016)

The first numerical simulations used to examine this effect followed the evolution of acold, dense core moving subsonically through a hot, magnetized ICM (Dursi 2007; Dursiand Pfrommer 2008; Asai et al. 2004, 2005, 2007), using a variety of field geometries.In all cases they confirmed the basic picture offered by Lyutikov (2006) of magnetic fieldamplification in a thin layer “draping” the cold front surface which forms at the head ofthe cool core. Figure 16 shows an example MHD simulation of a remnant-core cold frontproducing an amplified magnetic field in a draping layer from Asai et al. (2007).

A second type of cold front, the “sloshing” variety (Fig. 17), occurs in more relaxedsystems when the cold gas core is perturbed by infalling subclusters and is separated fromthe DM-dominated potential well. This gas then oscillates back and forth in the cluster cen-ter, producing a spiral-shaped pattern. Simulations have shown that sloshing cold frontsare also associated with amplified magnetic fields. The first simulations to demonstrate thiswere those of ZuHone et al. (2011), who simulated the evolution of initially tangled mag-netic fields with a number of initial magnetic field strengths and correlation scales. Thesloshing cold fronts are also associated with amplified magnetic layers, but unlike in thescenario envisaged by Lyutikov (2006) the layers are on the inside of the front surfacerather than outside, due to the fact that for the sloshing cold fronts the shear flow is pre-dominantly inside. This results in increased magnetic fields within the volume boundedby the cold fronts (Fig. 17), an effect important for the generation of radio mini-halos(Sect. 6.2).

6.1 Effects of Cold Front Magnetic Fields on the Thermal Plasma

The above considerations indicate that if a highly magnetized layer forms tangential to a coldfront or otherwise because of shearing motions that it may produce a dip in X-ray surfacebrightness at this location. These dips were first noticed in MHD simulations of sloshingcold fronts by ZuHone et al. (2011). In these simulations, the layers reached magnetic fieldstrengths with βpl ∼ 10 and dips in density and temperature of roughly ∼ 10–30%, whichcould produce dips in surface brightness of roughly ∼ 5–10%, depending on the gas temper-ature. The evidence for such features in X-ray observations of clusters is so far inconclusive,but there are some tantalizing hints (e.g. Werner et al. 2016) (Fig. 18).

The magnetic tension from a field stretched parallel to a front surface will suppress thegrowth of K-H instabilities if the field is strong enough (Chandrasekhar 1961). The ini-tial smooth appearance of many cold fronts as seen in Chandra observations led readily


Fig. 18 Evidence for amplified magnetic fields in X-ray observations of the Virgo cluster from Werner et al.(2016). Left panel: residual image of the X-ray surface brightness as seen by Chandra near the cold front.Three linear enhancements in surface brightness are apparent. Middle and right panels: Slices of gas densityand magnetic field strength near the cold front surface in a MHD simulation of the Virgo cluster. Wide bandsof strong magnetic field in between narrow channels of weak field produce linear features in density similarto those seen in Virgo

to the proposal that such suppression was occurring. For example, the apparent smooth-ness of the merger-remnant cold front in A3667 led Vikhlinin et al. (2001), Vikhlinin andMarkevitch (2002) to estimate a magnetic field strength near the front surface between6 μG < B < 14 μG. More recently, Chen et al. (2017) estimated a magnetic field strength ofB ∼ 20–30 μG at the sloshing cold fronts in A2204 based on the lack of observed KHI.

As deeper Chandra observations of nearby clusters with longer exposures have beenobtained over the years, some evidence for K-H instabilities has been uncovered. Roedigeret al. (2015a,b), Kraft et al. (2017) presented evidence of gas stripping of caused by KHI inthe elliptical galaxy M89 using deep X-ray observations and tailored simulations. Ichinoheet al. (2017) showed evidence for KHI in a longer combined exposure of A3667 than wasavailable to Vikhlinin et al. (2001), Vikhlinin and Markevitch (2002), but did not make anupdated estimate of the magnetic field strength. Finally, Su et al. (2017) showed evidencefor K-H instabilities at the interface of the cold front in NGC 1404, and used their presenceto place an upper limit on the magnetic field strength at the front of 5 μG.

However, the presence of some degree of KHI in cold fronts is not inconsistent with thepicture of magnetic draping layers per se—it is rather likely an indication of the strengthof the fields in these layers. A recent series of papers has constrained the magnetic fieldstrength in the Perseus cluster using MHD simulations and X-ray observations. Walker et al.(2017) showed convincing evidence of a giant KHI eddy at one of the cold front edges inthe Perseus cluster. They compared the appearance of the cold fronts to the simulationsfrom ZuHone et al. (2011), and suggested that a cluster with an initial β ∼ 200 before thesloshing began could explain the presence of the KHI eddy-simulations with initially largeror smaller average magnetic field strengths produced results that were inconsistent in termsof having either too few or too many KHI eddies along the interface.

6.2 Amplified Magnetic Fields and Cosmic Rays: Radio Mini-Halos

Radio mini-halos are the smaller-scale siblings of the giant radio halos, hosted in cool-coreclusters. Their emission is similarly diffuse and has a steep spectrum (α ∼ 1–2), but arenearly an order of magnitude smaller than radio halos and are confined to the core region.Mazzotta and Giacintucci (2008) were the first to discover that the radio mini-halos in theclusters RX J1720.1+2638 and MS 1455.0+2232 were confined to the region on the sky


Fig. 19 Association of radio mini-halos with cold fronts from ZuHone et al. (2013). Left panel: Chandraobservation of RXJ 1720.1+26, showing sloshing cold fronts, with 610 MHz radio contours overlaid. Theradio emission is coincident with the sloshing cold fronts. Middle panel: Simulated 327 MHz radio contoursoverlaid on projected temperature from an MHD/CRe simulation of sloshing core gas which produces amini-halo with emission bounded by the cold fronts. Right panel: Profiles of temperature and radio emissiontaken along the two directions shown in the middle panel, which show clearly that the radio emission dropssteeply at the position of the cold fronts

bounded by sloshing cold fronts seen in the X-ray observations. Subsequent investigationsof mini-halo emission from a number of cool-core clusters have confirmed the existenceof sharp drops in radio emission at the position of the cold front surfaces in many cases(Giacintucci et al. 2014a,b, 2017).

Such radio emission requires a population of CRe with γ ∼ 103–104, given the typicalmagnetic field strengths in clusters. Since CRe with such energies cool rapidly via syn-chrotron and Inverse-Compton losses, the existence of mini-halos requires a mechanismto replenish these electrons, either by reacceleration from a lower-energy population (the“reacceleration” model Brunetti and Lazarian 2007, 2011a,b) or as the byproducts of col-lisions of CRp with the ICM thermal proton population (the “hadronic” model, Dennison1980; Pfrommer and Enßlin 2004; Keshet and Loeb 2010), though this model is stronglyconstrained by the Fermi-LAT upper limits on gamma-ray emission in clusters (Ackermannet al. 2014), which are also produced by the same collisions. A review of these processes andtheir implications for non-thermal emission in clusters can be found in Brunetti and Jones(2014).

As previously noted, sloshing cold fronts are very common in cool-core clusters, andthese motions amplify magnetic fields. A stronger magnetic field within the core would leadto an enhancement of the mini-halo emission. Since this amplified magnetic field is largelyconfined to the volume bounded by the cold fronts, it may also explain the association ofmini-halos with cold fronts and the steep drops in radio emission coincident with the frontsurfaces, as suggested by Keshet and Loeb (2010).

The only simulations so far to directly test the reacceleration scenario for mini-haloswere performed by ZuHone et al. (2013). They used a MHD simulation of gas sloshing ina cool-core cluster coupled with a simulation of the evolution of the CRe spectrum underreacceleration by turbulence and radiative and Coulomb losses along trajectories of passivetracer particles. It was found that reacceleration by turbulence coupled with the magneticfield amplification, both produced by the sloshing motions, could produce mini-halos whichhave the characteristic diffuse emission, steep spectrum, and spatial relationship to the coldfronts (Fig. 19). They noted that the mini-halo produced in their simulation had two fur-ther interesting characteristics: the emission was transient and brightest shortly after thebeginning of the sloshing motions, and had a “patchy” appearance due to the intermittentand irregular distribution of turbulent gas motions in the core region. The latter prediction


is perhaps supported by the recent JVLA 230–470 MHz observations of the Perseus mini-halo by Gendron-Marsolais et al. (2017), which revealed a complex appearance of the radioemission.

The coincidence of mini-halos with cold fronts could also be explained by amplifiedmagnetic fields in the hadronic model. Keshet and Loeb (2010) also suggested that theserapidly amplifying magnetic fields may be responsible for the steep spectrum of mini-halos.ZuHone et al. (2015a) tested this possibility using simulations that the fast amplification ofmagnetic fields by sloshing motions could produce diffuse, core-confined mini-halos withsteep spectra by steepening the CRe spectrum. Though the amplification of the magneticfield strength within the sloshing cold fronts reproduced the observed spatial properties ofmini-halo emission in this simulation, they found that only a small, observationally insignifi-cant number of tracer particle trajectories experienced sufficiently rapid changes in magneticfield strength to steepen their radio spectra to α ∼ 2, and then only for brief periods of time.More complex mini-halo morphologies and spectral index properties may be produced inthe context of hadronic models by taking into account other CRp physics such as diffusion,streaming, and advection by turbulence and/or bulk motions (Enßlin et al. 2011; Pfrommer2013; Wiener et al. 2013, 2018; Jacob and Pfrommer 2017a,b).

7 Concluding Remarks

We have presented the most important mechanisms that are expected to control and drivethe amplification of magnetic fields observed in galaxy clusters at radio wavelengths. Wegave a short introduction to turbulence, motivated MHD as a model for the intra-cluster-medium and introduced the basic principles of the small-scale MHD turbulent dynamo. Wereviewed the outcome of (cosmological) numerical simulations of the growth of magneticfields, under typical conditions in the intracluster medium. We provided a short discussionof numerical limitations of current approaches and established the demands of upcomingradio surveys of Faraday rotation measures and giant radio haloes. We also introduced mag-netic field amplification at shocks and by cosmic-rays, which is evidenced by radio relicsat cluster outskirts. Further we discussed magnetic field amplification by cold fronts andinternal cluster motions, which are likely connected to radio mini halos.

The observed properties of magnetic fields in the intra-cluster medium require large am-plification factors (≥ 103) even considering the effect of gas compression. Indeed, there isplenty of time and turbulent energy to boost the magnetic field energy up to observed valueswith a turbulent small-scale dynamo. The theoretical grounds of this small-scale dynamomodel for the amplification of weak fields in random flows are robust and are covered in asignificant amount of literature. A central outcome of these studies is the importance of theAlfvén scale on the amplification and topology of magnetic fields in turbulent flows.

Given these expectations, the quantitative outcome of the small-scale dynamo in currentcosmological simulations is likely not sufficient to robustly predict magnetic fields in theICM for upcoming radio interferometers. Modern instruments require the robust predictionof magnetic field and shock structures down to a few kpc in the whole volume of a massivegalaxy cluster. This means future cosmological simulations will need to resolve the small-scale dynamo down to at least similar resolutions.

A way forward may be higher order MHD methods suitable for cosmology, that resolvemotions closer to the grid scale. Another possibility, especially for shocks, could be adaptivetechniques to selectively refine the mesh, where the amplification is active. We note that


Table 1 Reference parameters for the amplification of magnetic fields in various locations of the intra-cluster-medium. 2nd column: typical observed value; 3rd column: plasma beta; 4th column: inferred (mini-mum) amplification factor of magnetic field energy (also including gas compression); 5th column: amplifi-cation factor only considering dynamo amplification; 6th column: minimum amplification time; 7th column:typical scale of field reversals; 8th column: estimated growth factor in magnetic energy

Bobs βpl (δB/B0)2tot (δB/B0)2

dyn tdyn lB Γ

[μG] [Gyr] [kpc] [Gyr−1]

SSD ∼ 0.1–5 ∼ 102 ≥ 108 ≥ 103 ≤ 10 ≤ 50–100 ∼ 1

Shocks ∼ 1–3 ∼ 10 ≥ 100 ≥ 30 ≤ 0.01 ≤ 10–102 ∼ 0.003

Sloshing ∼ 10–20 ∼ 1 ≥ 10 ≥ 10 ≤ 0.01 ≤ 1–10 ∼ 0.001

ample computing power is available, as the largest computers approach 1018 floating pointoperations per second in the next years.

Given the large magnetic Reynolds number in the real intra-cluster medium, observed∼ μG fields can be the result of either primordial ∼ nG (co-moving) or ∼ 10−6 nG fields,as well as from higher and more concentrated magnetic seeds released by galactic winds oractive galactic nuclei. Density adaptive techniques are well suited for these kinds of sim-ulations. However, as opposed to galaxy formation, turbulence evolution and the SSD areonly weakly correlated with density peaks in the large-scale structure. Thus density adap-tive techniques might turn out to be inefficient for this problem, e.g. resolving shocks witha few kpc at the virial radius of a cluster would require exceedingly large particle num-bers.

The inferred amplification factors in clusters shocks traced by radio relics and at thecontact discontinuities generated by bulk motions in cluster cores are lower than from theSSD (∼ 10–30), yet amplification must again operate on small scales (≤ 102 kpc) and onshort timescales (∼ 10 Myr). Fully reproducing these trends with simulations may still be achallenge for numerical simulations, again because of resolution. Additionally, more com-plex interplay of magnetic fields and cosmic-rays are likely at work and plasma conditionsacross such discontinuities in the fluid may require a consideration of modified viscosity andthermal conduction.

In Table 1 we summarize these general trends in magnetic field amplification found fromobservations and guided by simulations, which we consider to be robust against numericalissues. We give the typically observed magnetic field, the estimated total magnetic energygrowth (considering for each case the magnetic field before the process begins; the mag-netic energy growth only due to small-scale dynamo (e.g. after removing for the ∝ n4/3

compression factor related to each process); the estimated (minimum) time for amplifica-tion; the typical energy containing scale related to each mechanism. The final column givesthe estimated growth factor, γgrowth for each amplification mechanism, parameterized by(δB/B0)

2dyn ≈ exp(Γ tdyn)).

In this review, we were addressing the successes and limitations of numerical models forpredictions of extragalactic magnetic fields. It seems obvious to foresee that the upcominggeneration of radio observations (culminating in the SKA data) will pose ever more chal-lenging questions to our theoretical and numerical models of rarefied space plasmas. Onlyin synergy will numerical techniques and radio observations exploit the next generation ofradio telescopes to study plasma physics in the Universe.

Acknowledgements The authors would like to thank the referee, D.C. Ellison, for valuable comments thatimproved the review. The authors thank K. Dolag for discussions on galactic dynamos. JD thanks T. Tricco,


D. Price, D. Ryu and T. Jones for discussions on numerical MHD methods. JAZ thanks C. Pfrommer foruseful comments.

F.V. acknowledges financial support from the ERC Starting Grant “MAGCOW”, no. 714196, and theusage of computational resources on the Piz-Daint supercluster at CSCS-ETHZ (Lugano, Switzerland) un-der project s701 and s805, and of the JURECA supercluster at JFZ (Jülich, Germany) under project hhh42(HiMAG). JD thanks H. Roettgering and Leiden observatory for the hospitality. This research has receivedfunding from the People Programme (Marie Sklodowska Curie Actions) of the European Unions EighthFramework Programme H2020 under REA grant agreement no. 658912, “CosmoPlasmas”.

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